One easily checks that \(W_{n}(T) = -1\)  whereas for every \(0 \leqslant i \leqslant n-1\) , we have \(W_{i}(T) \geqslant 0\)  as well as \(W_{i+1}(T)-W_{i}(T) \geqslant -1\) .
There is a 1-to-1 correspondence between the set of such paths and plane trees with \(n\)  vertices, see e.g. [1]} or [2]}. Furthermore the leaves of the tree correspond to the negative increments of its Łukasiewicz path. For every \(k \in \lbrace 0, \dots , n-1\rbrace \) , we shall denote by
\(\Lambda _{k}(T) = \sum _{i = 0}^{k} \mathbb {1}_{\lbrace W_{i+1}(T) - W_{i}(T) = -1\rbrace }\) 
